Linear homeomorphisms of some classical families of univalent functions

Abstract:

The extreme points of the closed convex hull of some classical families of univalent functions analytic on the open unit disk, e.g. the convex, K, and starlike, St, have recently been characterized. These characterizations are used to determine an explicit representation for the class of linear homeomorphisms of the extreme points of the closed convex hulls of K and St and thus of the hulls themselves. With the aid of these representations it is shown that every linear homeomorphism of K or St is a rotation, i.e. convolution with $\{ \exp (in \theta ):n = 0,1, \ldots \}$. In the way of a positive result: if $\mathcal {P}$ is the convex set of analytic functions with positive real part and $f(0) = 1$ and $\mathcal {L}$ is a linear homeomorphism of $\mathcal {P}$, then $\mathcal {L}(St) \subset St$, but $\mathcal {L}(K) \not \subset K$.